Show that the length of the subtangent at any point on the curve $y=a^x(a>0)$ is constant.

Let $P(x, y)$ be a point on the curve

$y = a^x$
diff. w.r.t to ‘$x$’

$\frac{dy}{dx}=a^x\log a$

slope $\left(m\right)={\left(\frac{dy}{dx}\right)}_{P(x,y)}=a^x\log a$

length of sub tangent $=\left|\frac{y}{m}\right|$

$=\left|\frac{{\mathcal{A}}^t}{\text{⧸}{A}^t}\log a\right|=\left|\frac{1}{\log a}\right|=$constant